Auslander studied the torsion-freeness of tensor powers of modules and obtained a criteria for freeness. More precisely, over an unramified regular local ring \(R\)  of dimension \(d\) , Auslander [1]} proved that an \(R\) -module \(M\)  is free provided that the \(d\) -fold tensor product \(M^{\otimes d}\)  of \(M\)  is torsion-free. The aim of this section is to generalize Auslander's result over more general rings, and find some new criteria for freeness of modules in terms of the local cohomology; see Corollary REF .
